I'm wondering how do Chief Justices of the United States get replaced? According to wikipedia, the last Chief Justice, Judge William Rehnquist was initially an Associate Justice. President Reagan nominated him to replace the retiring Chief Justice at the time, Judge Warren Burger.

When Judge Rehnquist passed away in 2005, Bush nominated someone outside of the Supreme Court, Judge John Roberts to become Chief Justice. I'm wondering, why someone on the current Supreme Court (at that time) wasn't nominated to become Chief Justice and an outsider was nominated? I would have assumed someone from the current Supreme Court (at that time) would have been nominated to become the new Chief Justice like Judge Rehnquist was and then nominate someone else to become the 9th Justice on the court.

2 Answers
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The position of Chief Justice contains some unique roles within SCOTUS, although these don't really affect the operations of SCOTUS directly (as in SCOTUS can operate completely without one, or in spite of one if need be). It's those special roles that require a dedicated confirmation process (i.e. an Associate must be nominated to Chief and be confirmed by the Senate). From Wikipedia

Despite the chief justice's elevated stature, his vote carries the same legal weight as the vote of each associate justice. Additionally, he has no legal authority to overrule the verdicts or interpretations of the other eight judges or tamper with them. The task of assigning who shall write the opinion for the majority falls to the most senior justice in the majority. Thus, when the chief justice is in the majority, he always assigns the opinion. Early in his tenure, Chief Justice John Marshall insisted upon holdings which the justices could unanimously back as a means to establish and build the Court's national prestige. In doing so, Marshall would often write the opinions himself, and actively discouraged dissenting opinions. Associate Justice William Johnson eventually persuaded Marshall and the rest of the Court to adopt its present practice: one justice writes an opinion for the majority, and the rest are free to write their own separate opinions or not, whether concurring or dissenting.

Bush had already nominated Roberts to Associate when Renquist died. He withdrew the nomination to renominate Roberts as Chief. It was a political decision, as there are no special qualifications to be Chief.

The Chief Justice of the United States isn't really different than any other Justice position. The only real difference in terms of nominations is that existing Justices are possible choices along with anyone else. As for why Bush chose an outsider for Chief Justice, the most likely reason was to pick someone that he agreed with and young enough to have a lasting impact. Choosing an existing Justice can be tricky because the have more publicized voting records and may not want the stress of facing another appointment.

The Justices at the time were Stevens(85), O'Connor(75), Scalia(69), Kennedy(69), Souter(66), Thomas(57), Ginsburg(72), and Breyer(67). Roberts at the time was 50, making him much younger than anyone else on the court at the time. Justices Ginsburg, and Breyer were nominated by Clinton and not likely considered. O'Connor already intended to retire, and Roberts was going to be her replacement. Stevens and Souter had liberal voting records so they were likely dismissed. This left Scalia, Kennedy, and Thomas as potential choices. Thomas faced an extremely tough first confirmation and voted very conservatively making his appoint likely to fail. Scalia was similarly conservative and would have faced similar difficulties getting confirmed. Kennedy was a potential option though he was slightly old, he also wasn't reliably conservative. Kennedy would have been an acceptable option, but the exact reason he wasn't chosen will likely remain speculation.

"The Chief Justice of the United States isn't really different than any other Justice position." The Chief Justice has many additional responsibilities like presiding over the administration of the federal judiciary (a cabinet sized department of the government with almost a thousand senior employees who can't be fired or have their pay cut for any reason), selecting judges to serve on the FISA court and court rules drafting committees, and presiding at Presidential impeachments.
– ohwillekeOct 17 '18 at 16:47